Geometry of the energy landscape of the self-gravitating ring

Abstract

We study the global geometry of the energy landscape of a simple model of a self-gravitating system, the self-gravitating ring (SGR). This is done by endowing the configuration space with a metric such that the dynamical trajectories are identified with geodesics. The average curvature and curvature fluctuations of the energy landscape are computed by means of Monte Carlo simulations and, when possible, of a mean-field method, showing that these global geometric quantities provide a clear geometric characterization of the collapse phase transition occurring in the SGR as the transition from a flat landscape at high energies to a landscape with mainly positive but fluctuating curvature in the collapsed phase. Moreover, curvature fluctuations show a maximum in correspondence with the energy of a possible further transition, occurring at lower energies than the collapsed one, whose existence had been previously conjectured on the basis of a local analysis of the energy landscape and whose effect on the usual thermodynamic quantities, if any, is extremely weak. We also estimate the largest Lyapunov exponent λ of the SGR using the geometric observables. The geometric estimate always gives the correct order of magnitude of λ and is also quantitatively correct at small energy densities and, in the limit N→∞, in the whole homogeneous phase.

N-body chaos and the continuum limit in numerical simulations of self-gravitating systems, revisited

“…These studies suggest a scaling of the form λmax ∝ N −1/3 or λmax ∝ 1/ ln N , depending on the total energy 3 of the model in the finite N regime; these results have been confirmed by Filho et al (2018). Similar scalings of λmax with N were found by Monechi & Casetti (2012) for the self-gravitating ring model, where particles interacting via softened gravitational forces are constrained on a ring. Using a differential geometry approach (see e.g.…”

Lyapunov Spectra of Coulombic and Gravitational Periodic Systems

“…When plotted versus temperature, this precursor transforms into a cusp characterized by a discontinuity of slope for the gravitational system in the thermodynamics limit [7]. Similar behavior was reported for a self-gravitating ring system [33]. However, as conjectured by Kunz [19], no such transitioning behavior is indicated for the Coulombic system.…”

“…LCEs represent the average rates of exponential divergence of nearby trajectories from a reference trajectory in different directions of the phase space and quantify the degree of chaos in a dynamical system [23][24][25][26][27]. In addition, LCEs have also been reported to serve as indicators of phase transitions [28][29][30][31][32][33].…”

Coarse-grained collisionless dynamics with long-range interactions

“…Instead of working as in Sec. IV A 3 with low-dimensional gravity, one can consider (softened) three-dimensional gravitational forces but constrain the interacting particles to move on a ring; the resulting model is referred to as the selfgravitating ring, introduced in [28] and further studied in [29][30][31][32]. The Hamiltonian, again expressed in dimensionless variables, is…”